Domination number of total graphs

Abstract

Abstract Let R be a commutative ring with Z(R) the set of zero-divisors and U(R) the set of unit elements of R. The total graph of R, denoted by T(Γ(R)), is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). We study the domination number of T(Γ(R)). It is shown that if R = Z(R) ∪ U(R), then the domination number of T(∪(R)) is finite provided R has a maximal ideal of finite index. Moreover, if R = ∏ i = 1 n F i $R = \prod\limits_{i = 1}^n {{F_i}} $ , where Fi is a field for each 1 ≤ i ≤ n and t = |F 1| ≤ |F 2| ≤ ··· ≤ |Fn |, then the domination number of T(Γ(R)) is equal to t - 1 provided t = |Fi | for every 1 ≤ i ≤ n, and is equal to t otherwise. Finally, for an R-module M it is shown that the total domination number of the total graph of the idealization (Nagata extension) R(+)M is equal to the domination number of the total graph of R provided M is a torsion free R-module or R = Z(R) ∪ U(R).